Difference between multiplying and dividing numbers by percentages?

Question

Basically, what is the difference between $1000\times1.03$ and $1000/.97$?

For some reason I feel like both should result in the same number. I only ask because I'm working a problem with a percentage of waste added in. For $3\%$ waste, I would think that you could multiply the amount by $1.03$ to add $3\%$. However, my professor divided by $.97$.

Answer 1

Just multiply

$$1.03 \times 0.97=0.9991$$ to see the difference.

Something that increases 3% and then decreases 3%... has slightly decreased. This is because the decrease of 3% is not measured compared to the original value, but to the increased value.

Answer 2

Well you see $1/.97\approx 1.0309$. So that’s actually about a 3.1% increase. Your professor was taking liberties with saying a 3% increase. That is $x/.97=1.031x$, slightly more than 3%. It is unknown whether the professor intentionally did something opposite of tradition.